Optimal Investment with Transaction Costs under Cumulative Prospect Theory in Discrete Time

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1 Optimal Investment with Transaction Costs under Cumulative Prospect Theory in Discrete Time Bin Zou and Rudi Zagst Chair of Mathematical Finance Technical University of Munich This Version: April 12, 2016 First Draft: October 14, 2015 Abstract We study optimal investment problems under the framework of cumulative prospective theory (CPT). A CPT investor makes investment decisions in a single-period financial market with transaction costs. The objective is to seek the optimal investment strategy that maximizes the prospect value of the investor s final wealth. We have obtained the optimal investment strategy explicitly in two examples. An economic analysis is conducted to investigate the impact of the transaction costs and risk aversion on the optimal investment strategy. Key Words: cumulative prospect theory; discrete time model; optimal investment; S-shaped utility; transaction costs. 1 Introduction In economics and finance, an essential problem is how to model people s preference over uncertain outcomes. To address this problem, Bernoulli (1954) bin.zou@tum.de. Phone: Corresponding address: Chair of Mathematical Finance, Technical University of Munich, Parkring 11, Garching 85748, Germany. zagst@tum.de. Phone:

2 (originally published in 1738) proposed expected utility theory (EUT): any uncertain outcome X is represented by a numerical value E[U(X)], which is the expected value of the utility U(X) taken under an objective probability measure P. An outcome X 1 is preferred to another outcome X 2 if and only if E[U(X 1 )] > E[U(X 2 )]. Hence, according to EUT, a rational individual seeks to maximize the expected utility E[U(X)] over all available outcomes. Bernoulli s original EUT was formally established by von Neumann and Morgenstern (thus the theory is also called von Neumann-Morgenstern utility theorem). Von Neumann and Morgenstern (1944) show that any individual whose behavior satisfies certain axioms has a utility function U and always prefers outcomes that maximize the expected utility. Since then, expected utility maximization has been one of the most widely used criteria for optimization problems concerning uncertainty, see, e.g., Merton (1969) and Samuelson (1969) for optimal investment problems. However, empirical experiments and research show that human behavior may violate the basic tenets of EUT, e.g., Allais paradox challenges the fundamental independence axiom of EUT. In addition, the utility function u( ) under EUT is concave, i.e. individuals are uniformly risk averse, which contradicts the risk seeking behavior in case of losses observed from behavioral experiments. Please refer to Kahneman and Tversky (1979) for many designed choice problems whose results cannot be explained by EUT. Alternative theories have been proposed to address the drawbacks of EUT, such as Prospect Theory by Kahneman and Tversky (1979), Rank-Dependent Utility by Quiggin (1982), and Cumulative Prospect Theory (CPT) by Tversky and Kahneman (1992). CPT can explain diminishing sensitivity, loss aversion, and different risk attitudes. Furthermore, unlike prospect theory, CPT does not violate the first-order stochastic dominance. The detailed characterizations of CPT are presented in Subsection 2.2. In this paper, we study optimal investment problems under the CPT framework including transaction cost. Jin and Zhou (2008) solve the problem in a continuous time model without transaction cost (Black-Scholes model) by splitting it into two Choquet optimization problems. The completeness of the market (the unique pricing kernel) plays an essential role in solving the problem in their paper, see also Carlier and Dana (2011). Optimal investment in a single-period discrete model under CPT, again without transaction cost, has been studied, e.g., by Bernard and Ghossoub (2010), He and Zhou (2011), Pirvu and Schulze (2012), and Carassus and Rásonyi (2015). Bernard and Ghossoub (2010) obtain an explicit optimal solution in a frictionless fi- 2

3 nancial market under the following assumptions: a piece-wise power utility, risk-free asset as the reference point and no short-selling constraint. They also study the properties of a new risk measure (called CPT-ratio in their paper) and conduct numerical simulations to investigate the impact of several factors, including mean, volatility, skewness, and risk aversion, on the optimal investment. He and Zhou (2011) consider the same problem as Bernard and Ghossoub (2010), but provide detailed analysis on the well-posedness of the problem by introducing a new measure of loss aversion (named largeloss aversion degree in their paper). They do not impose any constraint on the investment strategies and are able to find optimal solutions explicitly in two cases: (1) piece-wise power utility and risk-free asset as the reference point; (2) piece-wise linear utility and general reference point. Pirvu and Schulze (2012) generalize the previous work on this problem by considering a frictionless market consisting of one risk-free asset and multiple risky assets. Their main contribution is to provide a two-fund separation theorem between the risk-free asset and the market portfolio when the excess return has an elliptically symmetric distribution. Carassus and Rásonyi (2015) tackle the problem for the first time under a multi-period market model. They not only address the well-posedness issue of the problem but also establish the existence of optimal strategies under some assumptions. Without transaction costs, the optimal portfolio found in Merton s framework may lead to unrealistic strategies, e.g., buying stocks at infinite amount. In real life, transaction costs (bid-ask spread) are always present, albeit small for highly liquid assets. In their seminal paper, Magill and Constantinides (1976) claim through heuristic arguments that the optimal portfolio contains a no-trade region. Davis and Norman (1990) provide a rigorous treatment on optimal policies, solutions to the free boundary problem, and the value process (associated optimal expected utility). Shreve and Soner (1994) further generalize the results with viscosity techniques. Please refer to Kabanov and Safarian (2010) and the references therein for a comprehensive introduction and development on the mathematical theory of financial markets with transaction costs. The majority of existing literature on optimal investment problems with transaction costs, including those mentioned above, pursue an analysis for an investor who behaves according to EUT. In this paper, we consider optimal investment problems under the CPT framework in a discrete-time model with transaction costs, which, to our best knowledge, has not been studied before. The main contribution of our work is to obtain an explicit optimal investment strategy for a CPT investor under 3

4 two examples. The rest of the paper is organized as follows. Section 2 introduces the market model with transaction costs, and the three key components of the CPT framework. The main optimization problem is also formulated in Section 2. We review the utility function and the weighting function used in the literature regarding CPT in Section 3. We then obtain the optimal investment strategy in explicit form for two cases in Section 4 and Section 5, respectively. We provide an economic analysis in Section 6 to study how the optimal investment strategy is affected by transaction costs and risk aversion. The conclusions of our work are summarized in Section 7. 2 The Setup 2.1 The Financial Market with Transaction Costs We consider a single-period discrete-time financial market model equipped with a complete probability space (Ω, F, P). In the model, time 0 and time T (T > 0) represent present and future, respectively. The financial market consists of one risk-free asset and one risky asset (e.g., stock index). Trading the risk-free asset is frictionless. However, trading the risky asset will incur proportional transaction costs, and we denote such proportion by λ, where λ (0, 1). The risk-free return for the time period [0, T ] is r, where r 0 is a constant. That means if an investor deposits e 1 in the risk-free asset at time 0, he or she will receive e(1 + r) at time T. The (nominal) return on the risky asset is given by a random variable R. We assume the ask price of the risky asset S( ) is modeled by S(T ) = (1 + R) S(0), where S(0) is a positive constant. The bid price of the risky asset at time t is given by (1 λ)s(t), where t = 0, T. We assume F 0 is trivial and F T is the completion of σ(s(t )). Thus R is F T measurable. For any F T measurable random variable Z, we denote its cumulative distribute function (CDF) by F Z ( ) and the survival function by S Z ( ). By definition, S Z ( ) = 1 F Z ( ). We consider an investor with initial portfolio (x 0, y 0 ). That means the investor starts with x 0 and y 0 amount of money in the risk-free and the risky asset, respectively. The investor chooses the amount of money to be additionally invested in the risky asset at time 0, denoted by θ, and carried 4

5 out to terminal time T when it will be liquidated. Denote the investor s terminal wealth after liquidation by W (θ). We calculate W (θ) based on the value of θ. θ = 0 W (θ) = { (1 + r)x 0 + (1 + R)y 0, if y 0 < 0; (1 + r)x 0 + (1 λ)(1 + R)y 0, if y 0 0. θ > 0 and θ + y 0 < 0 θ > 0 and θ + y 0 0 W (θ) = (1 + r)x 0 + (1 + R)y 0 + (R r)θ. W (θ) = (1 + r)x 0 + (1 λ)(1 + R)y 0 + ( (1 λ)(1 + R) (1 + r) ) θ. θ < 0 and θ + y 0 < 0 W (θ) = (1 + r)x 0 + (1 + R)y 0 + ( 1 + R (1 λ)(1 + r) ) θ. θ < 0 and θ + y 0 0 W (θ) = (1 + r)x 0 + (1 λ)(1 + R)y 0 + (1 λ)(r r)θ. We can further write W (θ) in a universal form as ] W (θ) = (1 + r)(x 0 θ) + (1 + R)(y 0 + θ) λ [(1 + R)(y 0 + θ) + + (1 + r)θ, where x + := max{x, 0} and x := max{ x, 0} for all x R. In this financial market, the non-arbitrage condition reads as ( ) ( ) P (1 λ)(1 + R) < 1 + r > 0, and P 1 + R > (1 λ)(1 + r) > 0. (1) Remark 2.1. We ignore two degenerated cases: (i) (1 λ)(1 + R) 1 + r, and (ii) 1 +R (1 λ)(1 +r), under which the market is also arbitrage-free. If λ = 0, meaning the market is frictionless, then the non-arbitrage condition (1) simply reduces to 0 < P(R < r) < 1. 5

6 2.2 The CPT Framework Tversky and Kahneman (1992) propose cumulative prospect theory (CPT) as a performance criterion for decision making under uncertainty. A CPT model is characterized by the following three key features. Reference point Behavioral studies show that people do not evaluate final outcomes directly but rather compare them to some benchmark. In CPT, a reference point B is chosen to serve as the benchmark for evaluating an uncertain outcome. Let X denote the final wealth of an investment decision. If X B, X B is considered gains from the investment; if X < B, B X is viewed as losses. For example, if B is set to 0, then the terminology of gains and losses fits into the common language. Utility function Investors are not universally risk averse, instead they have different risk attitudes towards gains and losses. To fit those risk attitudes, CPT applies a S-shaped utility function, which consists of two different functions, u + and u, for gains and losses, respectively. The pathwise prospect utility of an investment strategy (with associated wealth X) is defined by u + (X(ω) B) 1 X(ω) B 0 u (B X(ω)) 1 X(ω) B<0, ω Ω, where 1 A is an indicator function of set A. Tversky and Kahneman (1992) propose a S-shaped function v (called value function) to evaluate the gains and losses, where v is continuous and increasing throughout R with v(0) = 0, concave in R + (for gains) but convex in R (for losses). In our setting, the function v is given by v(x) = u + (x) 1 x 0 u (x) 1 x<0, x R. We assume throughout this paper that u ± : R + R + are twice differentiable, strictly increasing, strictly concave and satisfy u ± (0) = 0. Weighting function 6

7 Investors tend to overweight extreme events (small probability events) but underweight normal events (large probability events). This behavior is captured in CPT by transforming objective cumulative probabilities into subjective cumulative probabilities using the weighting function (also called distortion function). The weighting function has a reverse S-shape, and two separate parts for gains and losses, denoted by w + ( ) and w ( ), respectively. We assume that w ± : [0, 1] [0, 1] are strictly increasing and differentiable, and satisfy w ± (0) = 0, and w ± (1) = 1. For a random wealth X, we define its positive prospect V + (X) and negative prospect V (X) by V + (X) := V (X) := B B The prospect utility of X is defined by Let D := X B, then we have V (X) = V + (X) V (X) = 0 u + (x B) d[ w + (S X (x))]; u (B x) d[w (F X (x))]. V (X) := V + (X) V (X). u + (x)d[ w + (S D (x))] 0 u ( x)d[w (F D (x))]. (2) Through integration by parts and change of variable, we rewrite V (X) in (2) as V (X) = 2.3 The Problem 0 w + ( SD (x) ) du + (x) 0 w ( FD ( x) ) du (x). (3) In the financial market described in Subsection 2.1, an investor selects investment strategy θ under the CPT framework introduced in Subsection 2.2. In other words, the investor wants to maximize the prospect utility V (W (θ)) 7

8 of his/her terminal wealth, which is defined by (2) or (3). The investor s terminal wealth W (θ) is a function of investment strategy θ, so is the prospect utility V (W (θ)). We then denote J(θ) := V (W (θ)). The reference point B can be random, and is given in the form of B = a (1 + r) + b (1 + R), where a, b R. (4) B is a linear combination of the future values of e 1 investment in the riskfree asset and the risky asset. If b = 0, then the reference point B is a fixed constant. In the above market setting, we implicitly assume that the investor we consider is a small investor, and his/her investment activities do not have any impact on the price of the risky asset. In addition, no investor in the real market has the capability to borrow the risk-free asset or short sell the risky asset at infinite amount. Those observations motivate us to constrain θ in a bounded region A, namely, θ A := [ θ m, θ M ], where both θ m, θ M 0. If the constrained optimal investment θ ( θ m, θ M ), then θ is also optimal to the unconstrained problem. If θ = θ m or θ M, then the CPT criterion gives the investor wrong incentive to pursue maximal prospect by taking infinite risk if θ m or θ M. Hence, imposing the constraint θ A helps the investor rule out the extreme decisions (θ = ± ), see Pirvu and Schulze (2012, Section 3.2). To make sure J(θ) is well defined, we need the following assumption. Assumption 2.1. We assume both V + (W (θ)) and V (W (θ)) are finite for all θ A. Proposition 2.1. Assumption 2.1 is satisfied if one of the following conditions holds. The risky return R is bounded, e.g., R is a discrete random variable and R. 8

9 The risky return R follows a normal distribution, log-normal distribution, or student-t distribution, and for x small enough, there exists some 0 < ɛ < 1 such that w ±(x) = O ( x ɛ), and w ±(1 x) = O ( x ɛ). Proof. The first result is obvious. For the proof of the second result, please refer to He and Zhou (2011, Proposition 1) and Pirvu and Schulze (2012, Proposition 2.1). We then formulate optimal investment problems with transaction costs under CPT as follows. Problem 2.1. In a financial market with transaction costs (as modeled in Subsection 2.1), an investor seeks the optimal investment strategy to maximize the prospect utility J(θ) of his/her terminal wealth. Equivalently, the investor seeks the maximizer θ to the problem J(θ ) = sup θ A J(θ) = sup θ A V (W (θ)). 3 Review on Utility Function and Weighting Function In a CPT model, excluding the reference point, there are two essential components: utility function and weighting function. We devote this section to the review of utility function and weighting function in the literature. Tversky and Kahneman (1992) choose the following piece-wise power utility function: u + (x) = x α, and u (x) = kx β, (5) where k > 0, 0 < α, β 1. The power utility function is the most commonly used one in optimal investment problems under CPT, see Barberis and Huang (2008), Bernard and Ghossoub (2010), He and Zhou (2011, Section 5.1), Pirvu and Schulze (2012, Section 4.1), and many others. The scaling property, u ± (θx) = θ α u ± (x), is the key to obtain explicit solutions. To make sure the utility function (5) satisfies all the conditions proposed in Subsection 2.2, we need to impose certain assumptions on the parameters α, β and k. 9

10 Assumption 3.1. (Power Utility Function) The utility function is given by (5) with the parameters satisfying the following conditions: 0 < α β < 1, and k > 1. Tversky and Kahneman (1992) estimate the values for the parameters as α = β = 0.88, and k = 2.25, which clearly satisfy the conditions imposed in Assumption 3.1. The power utility function is unbounded, and hence may lead to an illposed problem (either infinite CPT value or infinite optimal investment), see He and Zhou (2011) for detailed discussions. Another drawback of the power utility function is that it fails to explain high risk averse behavior, as pointed out in Rieger and Bui (2011). Apparently, a piece-wise exponential utility function does not encounter those drawbacks, see arguments in Köbberling and Wakker (2005). Exponential utility function has been used in CPT related optimization problems by Zou (2015), see also Pirvu and Schulze (2012, Section 4.3). Assumption 3.2. (Exponential Utility Function) The exponential utility function is given by where η +, η > 0, ζ > 1. u + (x) = 1 e η +x, and u (x) = ζ ( 1 e η x ), (6) He and Zhou (2011) claim that the concavity/convexity condition (imposed on the utility function) is insignificant, and hence can be ignored. They then propose linear utility function, see also Pirvu and Schulze (2012, Section 4.2), u + (x) = x, and u (x) = kx, k > 1. However, if an investor s preference is represented by a linear utility function, then the marginal utility is the same at any wealth level. According to linear utility preference, the pleasure of receiving 1 million is the same for a penniless investor and a billionaire, which clearly violates investor s behavior as captured by diminishing marginal utility. Hence, we argue that it is inappropriate to consider linear utility under the CPT framework, at least from the economic point of view. 10

11 The weighting function used in Tversky and Kahneman (1992) is given by x γ w + (x) = (x γ + (1 x) γ ), and w x δ (x) =. (7) 1/γ (x δ + (1 x) δ ) 1/δ As pointed out in Rieger and Wang (2006), the above weighting function may fail to be strictly increasing when γ, δ 0.25, but are indeed strictly increasing when γ, δ 0.5. The condition for strictly increasing weighting function is relaxed to γ, δ 0.28 in Barberis and Huang (2008). The estimated parameters are γ = 0.61 and δ = 0.69 in Tversky and Kahneman (1992), which satisfy the conditions for the weighting function introduced in Subsection 2.2. Prelec (1998) introduces the following weighting function w + (x) = e δ+ ( ln(x)) γ and w (x) = e δ ( ln(x)) γ, (8) where γ (0, 1) and δ +, δ > 0. Rieger and Wang (2006) use Perlec s weighting function with δ + = δ = 1. 4 Explicit Solution for Continuous Investors In this section, we consider the case in which the risky return R has a continuous distribution. 1 To obtain explicit solutions to Problem 2.1, we assume all the assumptions below hold in this section. Assumption The initial position on the risky asset is positive, y 0 > Short-selling is not allowed, θ m y 0, i.e., y 0 + θ 0, θ A. 3. The reference point B is given by B = (1 + r)x 0 + (1 λ)(1 + R)y 0. We take a = x 0 and b = (1 λ)y 0 in (4). 4. The utility function is of power type given by Assumption This is where the section name Continuous Investors comes from. 11

12 Remark 4.1. We make some comments on Assumption 4.1. Since y 0 > 0 and θ m y 0, investors are allowed to sell the risky asset (θ can be negative), but no more than what they currently own. However, the no short-selling constraint imposed in Bernard and Ghossoub (2010) is equivalent to θ 0. The case of y 0 < 0 is less interesting since the no short-selling constraint then implies θ > 0. For the given reference point B, we have B = W (0). That means the benchmark we select is the terminal wealth of the doing nothing strategy. First, due to W (0) = B, we obtain J(0) = 0. Next, we study two subproblems: (P1) sup J(θ), and (P2) sup 0 θ θ M θ m θ 0 J(θ). By comparing the optimal prospect utility of the two sub-problems (P1) and (P2), we obtain the solution to Problem Solution to Sub-Problem (P1) If θ 0, then y 0 + θ 0. Hence the investor needs to sell all the holdings in the risky asset at liquidation. Define random variable Z 1 := (1 λ)(1 + R) (1 + r), then we obtain Define set A 1 by D = W (θ) B = Z 1 θ. A 1 := {Z 1 < 0} = { 1 + R < 1 + r }. 1 λ Notice that set A 1 is the set of losses for the investor (recall θ 0). Due to the non-arbitrage condition (1), P(A 1 ) > 0. By definition (2) and change of variable (x = zθ, θ > 0), the prospect utility J(θ) in sub-problem (P1) is obtained by J(θ) = = 0 0 x α d[ w + (S D (x))] 0 z α d [ w + ( SZ1 (z) )] θ α 12 k( x) β d[w (F D (x))] 0 ( z) β d [ w ( FZ1 (z) )] kθ β.

13 We define, for any F T measurable random variable Z, that g 1 (Z) : = l 1 (Z) : = 0 0 z α d [ w + ( SZ (z) )], ( z) β d [ w ( FZ (z) )]. (9) In general, g 1 (Z) (or l 1 (Z)) can be understood as the prospect value of gains (or losses) of random wealth X with X B = Z. In our setting here, g 1 (Z 1 ) and l 1 (Z 1 ) are exactly the prospect value of gains and the prospect value of losses (differ by a scalar k) of W (1), which is the terminal wealth associated with the strategy θ = 1. Mathematically, we have g 1 (Z 1 ) = V + (W (1)) and k l 1 (Z 1 ) = V (W (1)), and hence both are finite due to Assumption 2.1. With the definitions of g 1 and l 1, the prospect utility J(θ) is simplified as Define K 1 (Z 1 ) by J(θ) = g 1 (Z 1 ) θ α l 1 (Z 1 ) kθ β, θ 0. K 1 (Z 1 ) := g 1(Z 1 ) l 1 (Z 1 ). Given Assumption 2.1, K 1 (Z 1 ) is well defined and K 1 (Z 1 ) > 0. Notice that K 1 (Z) shares similar features of the Omega Measure proposed by Keating and Shadwick (2002), see Bernard and Ghossoub (2010, Section 4.1) for comparisons. We summarize the solution to sub-problem (P1) below. Theorem 4.1. If Assumption 2.1 and Assumption 4.1 hold, then the optimal investment θ to sub-problem (P1) is obtained from one of the following scenarios. 1. If P(A 1 ) = P(Z 1 < 0) = 1, then θ =0. 2. Let 0 < P(A 1 ) < 1. We obtain (a) If α = β and k > max{1, K 1 (Z 1 )}, then θ =0. (b) If α = β and k = K 1 (Z 1 ) > 1, then θ = [0, θ M ]. That means any θ [0, θ M ] is optimal. (c) If α = β and 1 < k < K 1 (Z 1 ), then θ = θ M. 13

14 (d) If α < β, then θ = Θ 1 := min{θ 1, θ M }, with θ 1 defined by θ 1 := ( ) 1 α βk K β α 1(Z 1 ). (10) Proof. If P(A 1 ) = 1, then the probability of suffering losses is 1 for all long strategies θ 0. Thus it is never optimal to buy the risky asset, i.e., θ = 0. Mathematically, P(A 1 ) = P(Z 1 < 0) = 1 g 1 (Z 1 ) = 0. Then we have J(θ) = l 1 (Z 1 ) kθ β < J(0) = 0, for all θ > 0, which directly indicates θ = 0. We next consider the non-trivial case: 0 < P(A 1 ) < 1. Differentiating J(θ) gives J (θ) = l 1 (Z 1 ) θ α 1 [ α K 1 (Z 1 ) βk θ β α]. If α = β, depending on the value of k, J(θ) is either a strictly decreasing or strictly increasing function or a constant, as summarized in (2a)-(2c). In Scenario (2c), J(θ) is strictly increasing, and lim θ J(θ) = + (called illposed case in He and Zhou (2011)). So the constraint is binding, and we have θ = θ M. If α < β, then θ 1, defined by (10), is the unique solution to J (θ) = 0 on the positive axis. Furthermore, J (θ) > 0 for all θ (0, θ 1 ) and J (θ) < 0 for all θ (θ 1, ). Therefore, θ 1 is the unique maximizer to the problem sup θ 0 J(θ). With the constraint θ θ M, the optimal investment θ = min{θ 1, θ M } := Θ Solution to Sub-Problem (P2) Due to the no short-selling constraint θ m y 0, we have y 0 + θ 0 for all θ m θ 0. So the liquidation order at terminal time T is to sell all the risky assets. Define Z 2 := (1 λ)(r r). We obtain in this case that D = W (θ) B = Z 2 θ. Define set A 2 by A 2 := {Z 2 > 0} = {R > r}. Since investment θ is restricted to short strategies (θ 0) in this subsection, the difference D is negative on set A 2, meaning that set A 2 is the set of losses for the investor. 14

15 In this case, we obtain J(θ) as follows: J(θ) = = 0 0 x α d[ w + (S D (x))] 0 ( z) α d [ w + ( FZ2 (z) )] ( θ) α k( x) β d[w (F D (x))] 0 z β d [ w ( SZ2 (z) )] k( θ) β, where we have applied the change of variable x = zθ (θ < 0) in the second equality. We define, for any F T measurable random variable Z, that g 2 (Z) : = l 2 (Z) : = 0 0 ( z) α d [ w + ( FZ (z) )], (z) β d [ w ( SZ (z) )]. (11) The economic meanings of g 2 (Z) and l 2 (Z) are similar to those of g 1 (Z) and l 1 (Z), except the gains/losses are located on exactly opposite tails due to the different signs of θ in two cases. Using the notations of g 2 ( ) and l 2 ( ), we rewrite J(θ) as J(θ) = g 2 (Z 2 ) ( θ) α l 2 (Z 2 ) k( θ) β, for y 0 θ 0, which is well defined if Assumption 2.1 holds. The unique solution to J (θ) = 0 on the negative axis is given by ( ) 1 α θ 2 := βk K β α 2(Z 2 ), where K2 (Z 2 ) := g 2(Z 2 ) l 2 (Z 2 ) > 0. (12) We directly provide the results to sub-problem (P2). Theorem 4.1 for a similar proof. Please refer to Theorem 4.2. If Assumption 2.1 and Assumption 4.1 hold, then the optimal investment θ to sub-problem (P2) is obtained from one of the following scenarios. 1. If P(A 2 ) = 1, then θ = If P(A 2 ) = 0, then θ = θ m. 15

16 3. 0 < P(A 2 ) < 1 (a) If α = β and k > max{1, K 2 (Z 2 )}, then θ = 0. (b) If α = β and k = K 2 (Z 2 ) > 1, then θ = [ θ m, 0]. (c) If α = β and 1 < k < K 2 (Z 2 ), then θ = θ m. (d) If α < β, then θ = Θ 2 := max{θ 2, θ m }. 4.3 Main Results To find the optimal solution to Problem 2.1, we compare the optimal prospect utility obtained from sub-problems (P1)-(P2) for different scenarios. Denote K M = max{k 1 (Z 1 ), K 2 (Z 2 )}. The main results are summarized in the theorem below. Theorem 4.3. If Assumption 2.1 and Assumption 4.1 hold, we have the following results for the optimal investment θ to Problem If P(A 1 ) = 1 and P(A 2 ) = 1, then θ = If P(A 1 ) = 1 and P(A 2 ) = 0, then θ = θ m. 3. If P(A 1 ) = 1 and 0 < P(A 2 ) < 1, θ is given by Case 3 of Theorem If 0 < P(A 1 ) < 1 and P(A 2 ) = 1, θ is given by Case 2 of Theorem < P(A 1 ) < 1 and 0 < P(A 2 ) < 1 (a) If α = β and k > max{1, K M }, then θ = 0. (b) If α = β and k = K 1 (Z 1 ) > max{1, K 2 (Z 2 )}, then θ = [0, θ M ]. (c) If α = β and k = K 2 (Z 2 ) > max{1, K 1 (Z 1 )}, then θ = [ θ m, 0]. (d) If α = β and k = K 1 (Z 1 ) = K 2 (Z 2 ) > 1, then θ = [ θ m, θ M ]. (e) If α = β and 1 < k < K M, then θ = arg max J(θ). { θ m, θ M } (f) If α < β, then θ = arg max J(θ). {Θ 1, Θ 2 } Proof. We only remark that P(A 2 ) = 0 P(A 1 ) = 1. All the results in the theorem are immediate consequences of Theorem 4.1 and Theorem

17 Remark 4.2. In Scenario 5(f), if the constraint is not binding, namely, Θ 1 = θ 1 and Θ 2 = θ 2, then we obtain finer results: (i) If α < β and (g 1 (Z 1 )) β /(l 1 (Z 1 )) α (g 2 (Z 2 )) β /(l 2 (Z 2 )) α, then θ = θ 1. (ii) If α < β and (g 1 (Z 1 )) β /(l 1 (Z 1 )) α < (g 2 (Z 2 )) β /(l 2 (Z 2 )) α, then θ = θ 2. The above results are based on the comparison between J(θ 1 ) and J(θ 2 ), see He and Zhou (2011, Appendix). Using the CPT definition (3), we rewrite g i (Z j ), i, j = 1, 2, as g 1 (Z 1 ) = g 2 (Z 2 ) = 0 0 w + (S Z1 (z))du + (z), w + (F Z2 ( z))du + (z), l 1 (Z 1 ) = 1 k l 2 (Z 2 ) = 1 k 0 0 w (F Z1 ( z))du (z), w (S Z2 (z))du (z). We have Z 2 > Z 1 almost surely, which implies F Z2 (z) F Z1 (z) for all z (strict inequality holds for some z). Furthermore, if Z 2 is symmetrically distributed around 0 (equivalently, R is symmetrically distributed around r), we have, z > 0 F Z2 ( z) = 1 F Z2 (z) 1 F Z1 (z) = S Z1 (z), F Z1 ( z) = 1 S Z1 ( z) 1 S Z2 ( z) = S Z2 (z). Therefore g 2 (Z 2 ) > g 1 (Z 1 ) and l 2 (Z 2 ) < l 1 (Z 1 ). Consequently, K 2 (Z 2 ) > K 1 (Z 1 ) holds, and then K M = K 2 (Z 2 ). Comparing Theorem 4.3 with the results in a frictionless market (see, for instance, Bernard and Ghossoub (2010, Theorem 3.1) and He and Zhou (2011, Theorem 3)), there are several differences: if λ = 0, then 0 < P(A 1 ) < 1 and 0 < P(A 2 ) < 1, so Cases (1)-(4) in Theorem 4.3 will never happen. if λ = 0, then Z 1 = Z 2 = R r. However, with λ > 0, we have Z 1 < Z 2 < R r. If Z 2 is symmetrically distributed around 0, we have K 1 (Z 1 ) = K 2 (Z 2 ) if λ = 0, but K 1 (Z 1 ) < K 2 (Z 2 ) if λ > 0. 17

18 4.4 Discussions for y 0 = 0 If the investor does not hold any risky asset at time 0 (y 0 = 0), we can remove the constraint of no short-selling and still obtain explicit solutions. Notice that B = (1 + r)x 0 when y 0 = 0, which is the most common choice for the reference point and is used by Bernard and Ghossoub (2010), He and Zhou (2011), Pirvu and Schulze (2012), and many others. The solution to sub-problem (P1) is exactly the same as in Theorem 4.1. However, the solution to sub-problem (P2) here is different from the results in Theorem 4.2. Since y 0 = 0, we have y 0 + θ 0 for all θ [ θ m, 0] (recall y 0 + θ 0 in Subsection 4.2). Define Z 3 := 1 + R (1 λ)(1 + r). Then we have Define the set of losses A 3 by D = W (θ) B = Z 3 θ for all θ [ θ m, 0]. A 3 := {Z 3 > 0} = {1 + R > (1 λ)(1 + r)}. The non-arbitrage condition (1) implies P(A 3 ) > 0, while P(A 2 ) = 0 is possible in Subsection 4.2. By replacing Z 2 by Z 3, A 2 by A 3 and removing the scenario of P(A 2 ) = 0 in Theorem 4.3, we obtain the optimal investment strategy to Problem 2.1 in the case of y 0 = 0. To study the connection between those two cases, we modify the notations for J(θ). For initial position (x 0, y 0 ) with y 0 > 0, we use J(θ, x 0, y 0 ) instead of J(θ). For initial position x 0 and y 0 = 0, we use J(θ, x 0 ) instead of J(θ). If an investor has an initial wealth x 0 + y 0, then he/she can buy the risky asset of amount y 0 and hold portfolio (x 0, y 0 ). On the other hand, if an investor holds portfolio (x 0, y 0 ) at the beginning, he/she can liquidate the risky asset and deposit all the money, x 0 + (1 λ)y 0, in the risk-free asset. Therefore, we obtain J( θ 1, x 0 + (1 λ)y 0 ) J(θ, x 0, y 0 ) J( θ 2, x 0 + y 0 ), where θ 1, θ, and θ 2 are the optimal investment strategies to the corresponding initial position. 18

19 5 Explicit Solution for Binomial Investors In this section, we consider a binomial market 2 specified by { u, with probability 1 p 1 + R =, (13) d, with probability p where u > d > 0 and 0 < p < 1. The non-arbitrage condition (1) in this model reads as u > (1 λ)(1 + r) > (1 λ) 2 d. Given a payoff ξ F T, we have { ξ u, when 1 + R = u; ξ = ξ d, when 1 + R = d. In what follows, we may denote ξ = (ξ u, ξ d ) in the above sense. In the market modeled by (13), assume we can replicate ξ by strategy θ ξ and initial investment x ξ. If ξ u ξ d, then we obtain 3 where p b u and p b d are defined by p b u := If ξ u < ξ d, then we obtain 4 ξ u ξ d θ ξ = (1 λ)(u d), (14) x ξ = 1 ( ) p b 1 + r u ξ u + p b d ξ d, (15) (1 + r) (1 λ)d, p b d := (1 λ)(u d) (1 λ)u (1 + r). (1 λ)(u d) θ ξ = ξ u ξ d u d, (16) x ξ = r (ps u ξ u + p s d ξ d ), (17) 2 The binomial distribution of the risk return suggests the name binomial investors. 3 In this case, the replication strategy involves long the risky asset. θ ξ and x ξ are solved from (1 + r) (x ξ θ ξ ) + (1 λ)u θ ξ = ξ u and (1 + r) (x ξ θ ξ ) + (1 λ)d θ ξ = ξ d. 4 In this case, the replication strategy involves short the risky asset. θ ξ and x ξ are solved from (1 + r) (x ξ (1 λ)θ ξ ) + u θ ξ = ξ u and (1 + r) (x ξ (1 λ)θ ξ ) + d θ ξ = ξ d. 19

20 where p s u and p s d are defined by p s u := (1 λ)(1 + r) d, p s d := u d u (1 λ)(1 + r). u d Remark 5.1. If ξ u ξ d (or ξ u < ξ d ), the replication strategy involves buying (or selling) the risky asset (since θ ξ 0 in (14) and θ ξ < 0 in (16)). Notice that p b u + p b d = 1, ps u + p s d = 1 and pb u, p s d > 0, but pb d and ps u may be negative, so (p b u, p b d ) and (ps u, p s d ) are not necessarily risk-neutral probability measures. However, if λ = 0, we have p b u = p s u, p b d = ps d, and (pb u, p b d ) is indeed the unique risk-neutral probability measure. Since we impose a trading constraint θ A = [ θ m, θ M ], the replication strategy θ ξ, given by (14) or (16), may not be attainable under the constraint. In this section, we shall study Problem 2.1 without constraint and let the unconstrained solution suggest whether the constraint is binding. To solve Problem 2.1, we claim that the assumptions below hold in the rest of this section. Assumption The investor begins with initial portfolio (x 0, 0), i.e., the investor does not hold any risky asset at the beginning, y 0 = The risky return in the market is modeled by (13). 3. The reference point is given by B = (1 + r)x The utility function is of exponential type as given by (6) in Assumption 3.2 with η + = η = η. Remark 5.2. Since y 0 = 0, we set x ξ = x 0 and only consider investment strategies with initial wealth x 0. Note that J(θ ξ ) = V (W (θ ξ )) = V (ξ) by (14) and (16). The assumption on the reference point together with (15) and (17) give that B = p b u ξ u + p b d ξ d = p s u ξ u + p s d ξ d. 20

21 Due to the above remarks, we consider two sets of random payoffs: Ξ b := {ξ = (ξ u, ξ d ) F T : ξ u ξ d, p b u (ξ u B) + p b d (ξ d B) = 0}, Ξ s := {ξ = (ξ u, ξ d ) F T : ξ u < ξ d, p s u (ξ u B) + p s d (ξ d B) = 0}. To solve the maximization problem sup θ R J(θ), we consider two sub problems: (P3) sup V (ξ) and (P4) sup V (ξ). ξ Ξ b ξ Ξ s 5.1 Solution to Sub-Problem (P3) If p b 1+r d 0, i.e., u < (corresponding to P(A 1 λ 1) = 1 in Subsection 4.1), then V (ξ) 0 = V ((B, B)). Hence ξ = (B, B) Ξ b, and θ = 0 because of (14). If p b d (0, 1), we immediately have ξ d B 0 ξ u B. ξ Ξ b, B ξ d = pb u (ξ p b u B). By the definition of CPT, we write V (ξ) as d V (ξ) = w + (1 p) u + (ξ u B) w (p) u (B ξ d ) ( ) p b = w + (1 p) u + (ξ u B) w (p) u u (ξ p b u B) d Then sub-problem (P3) is equivalent to sup ξu B L b (ξ u ). := L b (ξ u ). p b u = p b d In this case, using Assumption 3.2, we rewrite L b (ξ u ) as Define a threshold ζ by L b (ξ u ) = ( w + (1 p) ζw (p) ) u + (ξ u B). ζ := pb d w +(1 p). p b u w (p) Notice ζ = w + (1 p)/w (p) when p b u = p b d. Therefore, we obtain the optimal payoff ξu by B, when ζ > ζ ξu = [B, + ), when ζ = ζ > 1. +, when 1 < ζ < ζ 21

22 Here, the condition ζ > 1 comes from Assumption 3.2 (so called loss aversion condition). Hence, using (14) and ξ d = 2B ξ u, the optimal investment θ in [0, θ M ] is given by 0, when ζ > ζ θ = [0, θ M ], when ζ = ζ > 1. (18) θ M, when 1 < ζ < ζ p b u > p b d We calculate (L b ) (ξ u ) as (L b ) (ξ u ) = w + (1 p) ηe η(ξu B) ( 1 ζ ζ e η(pb u/p b d 1)(ξu B) ). If ζ ζ, then (L b ) (ξ u ) > 0 for all ξ u > B. The prospect L b (ξ u ) is a strictly increasing function of ξ u (and thus θ), hence the optimal investment in [0, θ M ] is θ = θ M. If ζ > ζ, we obtain (L b ) (ξu) = 0 ξu = B + ln ( ζ/ ζ ) ( ) > B, η p b u 1 p b d (L b ) (ξ u ) 0 ξ u ξ u, i.e., ξ u is minimum. The constraint θ [0, θ M ] is equivalent to ξ u [B, ξ M ], where ξ M defined through ξ M B θ M = (1 λ)(u d). Hence, if ζ > ζ, we have p b d is sup J(θ) = max{l b (B) = 0, L b (ξ M )}. θ [0, θ M ] To summarize, if p b u > p b d, the optimal payoff ξ u in [B, ξ M ] is given by ξu = arg max{l b (B), L b (ξ M )}, {B, ξ M } 22

23 and the optimal investment θ in [0, θ M ] is obtained as { θ 0, if ξu = B =. (19) θ M, if ξu = ξ M p b u < p b d Due to the analysis above, we easily obtain that (L b ) (ξ u ) < 0 when ζ ζ, and thus θ = 0 in this scenario. If ζ < ζ, solving (L b ) (ξ u ) = 0 gives ( ξu p b d ζ = B + η(p b d pb u) ln ζ and then θ 3 := = ) > B, ) ( ζ/ζ ξu B ln = p b d (1 λ)(u d) η(1 λ)(u d)(p b d ( pb u) 1 ζ η((1 λ)(u + d) 2(1 + r)) ln ζ ). (20) Since (L b ) (ξ u ) 0 ξ u ξu, ξu is the unique maximizer to the problem sup ξu B L b (ξ u ). Notice that θ 3 > 0 since p b u < p b d and ζ < ζ. Therefore, if p b u < p b d, the optimal investment θ in [0, θ M ] is { θ 0, if ζ = ζ. (21) Θ 3 := min{θ 3, θ M }, if ζ < ζ 5.2 Solution to Sub-Problem (P4) If p s u 0, then V (ξ) 0 for all ξ Ξ s, hence θ = 0. If p s u > 0, then ξ Ξ s, we have ξ u B 0 ξ d B, and B ξ u = p s d p s d (ξ d B). Hence, V (ξ) = w + (p) u + (ξ d B) w (1 p) u (B ξ u ) ( ) p s = w + (p) u + (ξ d B) w (1 p) u d (ξ p s d B) u 23 := L s (ξ d ).

24 The first derivative of L s (ξ d ) is calculated as [ ] (L s ) (ξ d ) = ηe η(ξd B) w + (p) 1 ζ ζ e η ( p s d )(ξ p s 1 d B) u, where constant ζ is defined by By (16), we derive ζ := ps u w + (p) p s d w (1 p). θ = ξ d B p s u(u d). It is obvious that sub-problem (P4) and sup ξd B L s (ξ d ) are equivalent. The analysis is the same as for sup ξu B L b (ξ u ) in the previous subsection, and we summarize results below. p s u = p s d In this case, we have sign ( (L s ) (ξ d ) ) = sign ( 1 ζ ) = sign (J (θ)), ζ and then 0, when ζ > ζ θ = [ θ m, 0], when ζ = ζ > 1. (22) θ m, when 1 < ζ < ζ p s u < p s d If ζ ζ, then (L s ) (ξ d ) > 0 for all ξ d > B and J (θ) < 0 for all θ < 0. Hence θ = θ m. If ζ > ζ, we obtain (L s ) (ξ d) = 0 ξ d = B + ( p s u ζ η(p s u p s d ) ln ζ (L s ) (ξ d ) 0 ξ d ξ d, i.e., ξ d is minimum. ) > B, 24

25 Therefore, the maximum will be achieved at the end points, i.e., sup J(θ) = max{j(0) = 0, J( θ m )}. θ [ θ m, 0] In conclusion, if p s u < p s d, the optimal investment θ is given by θ = arg max J(θ). (23) { θ m, 0} p s u > p s d If ζ ζ, then (L s ) (ξ d ) < 0 for all ξ d > B and J (θ) > 0 for all θ < 0, so θ = 0. If ζ < ζ, we obtain (L s ) (ξ d ) 0 ξ d ξ d, which implies that ξ d = arg max Ls (ξ d ). For (ξu, ξd ), where ξ u satisfies p s u (ξu B) + p s d (ξ d B) = 0, the corresponding replication strategy is given by ( ) ln ζ/ζ θ 4 = ξ u ξd u d = ξ d B p s u(u d) = η(u d)(p s u p s d ( ) 1 ζ = η ( 2(1 λ)(1 + r) (u + d) ) ln ζ ). (24) We then obtain the optimal investment in [ θ m, 0] { θ 0, if ζ ζ = Θ 4 := max{θ 4, θ m }, if ζ < ζ. (25) 5.3 Main Results To obtain an explicit solution to Problem 2.1, it remains to compare the optimal prospect utility obtained in the previous two subsections case by case. A major difference between the power utility (Assumption 3.1) and the exponential utility (Assumption 3.2) is that prospect utility under the exponential utility is always finite, due to the fact that 0 u ± (x) ζ for all x 0. Hence Assumption 2.1 is always satisfied under the exponential utility. 25

26 Theorem 5.1. Let Assumption 5.1 hold, we then obtain the optimal investment strategy θ to Problem 2.1 through the following cases. 1. If p b d 0, then sup θ [ θ m, θ M ] J(θ) = sup θ [ θm, 0] J(θ), and θ is given by (22), (23), or (25) depending on the comparison of p s u and p s d. 2. If p s u 0, then sup θ [ θm, θ M ] J(θ) = sup θ [0, θm ] J(θ), and θ is given by (18), (19), or (21) depending on the comparison of p b u and p b d. 3. If p b d > 0 and ps u > 0, or equivalently d separate the discussions as follows. 1 λ < 1+r < (1 λ)u, we further (a) p b u = p b d ( ps u < p s d ) { J(θ 0, if ζ ) = ζ, ζ > ζ, J( θ m ) < 0; max{j( θ m ), J(θ M )}, otherwise. (b) p s u = p s d ( pb u > p b d ) { J(θ 0, if ζ > ) = ζ, ζ ζ, J(θ M ) < 0; max{j( θ m ), J(θ M )}, otherwise. (c) p b u > p b d and ps u < p s d J(θ ) = max{j( θ m ), J(0), J(θ M )}. (d) p b u > p b d and ps u > p s d J(θ ) = max{j(θ 4 ), J(0), J(θ M )}. (e) p b u < p b d and ps u < p s d J(θ ) = max{j( θ m ), J(0), J(Θ 3 )}. If λ λ := max{1 1+r, 1 d }, both u 1+r pb d, ps u 0, then by Theorem 5.1, the optimal investment θ = 0. This result shows the optimal investment largely depends on transaction costs. CPT investors will not trade the risky asset as long as λ is above the threshold λ. However, if there are no transaction costs in the market (λ = 0), then the non-arbitrage condition d < 1 + r < u implies that λ > λ = 0. 26

27 6 Economic Analysis In this section, we conduct an economic analysis to study how the optimal investment strategy is affected by transaction costs and risk aversion. The calculations in Section 5 are straightforward as long as the binomial model (13) has been estimated. However, under Tversky and Kahneman s weighting functions (7) or Prelect s weighting functions (8), the numerical calculations for K 1 (Z 1 ) and K 2 (Z 2 ) (two integrals) in Section 4 are very complicated even when the risky return 1 + R is normally distributed or lognomarlly distributed. In what follows, we obtain numerical results based on the model in Section 4 and conclusions from Theorem Data and Model Parameters We consider optimal investment problems in a single-period discrete model, so we select a relatively short time window. In the economic analysis thereafter, we select the time window to be 1 week, T = 1 week. To estimate the risk-free interest rate r, we use 3-month EONIA (Euro OverNight Index Average) Swap Index bid close quotes between January 2, 2012 and June 30, There are 891 daily observations during the selected time period. 5 To have more consistent data, we convert the daily frequency into weekly. The descriptive statistics for the weekly quotes are summarized in the table below. Obs. Mean Median Std. Skewness % % Table 1: Summary Statistics of Annualized Risk-free Return Due to the right skewness, we choose the median as the estimate for the risk-free return. Then the weekly risk-free return r is obtained by r = ( %) 1/52 1 = In order to estimate the distribution of the risky return R, we choose the weekly close quotes of FTSE (Financial Times Stock Exchange) 100 Index 5 Data source: Thomson Reuters Eikon. Access from the Chair of Mathematical Finance at the Technical University of Munich is greatly appreciated. 27

28 from January 2, 2012 to July 6, FTSE 100 index and obtain We calculate the log return of the µ = , and σ = The QQ plot of ln(1 + R) versus standard normal in Figure 1 suggests that Quantiles of Input Sample Standard Normal Quantiles Figure 1: QQ Plot of ln(1 + R) versus Standard Normal ln(1 + R) is approximately normal. From now on, we assume ln(1 + R) N(µ, σ 2 ). For the numerical calculations in this section, we select Tversky and Kahneman s weighting function (given by (7)) with parameters γ = 0.61 and δ = We consider power utility function as in Assumption 3.1. The risk attitudes of an CPT investor depend on α and β. We separate the discussions into two cases: α = β and α < β. 6 Data source: Yahoo Finance 28

29 6.2 The Case of α = β If α = β, the optimal investment strategy is given by one of the scenarios (2a), (2b), or (3a)-(3e) in Theorem 4.3. In the analysis, we select α = β = 0.88, as estimated in Tversky and Kahneman (1992). Since ln(1 + R) is normally distributed, we have 0 < P(A 1 ), P(A 2 ) < 1. Then according to Case (3) in Theorem 4.3, we need to calculate K 1 (Z 1 ) and K 2 (Z 2 ) in order to obtain the optimal investment strategy θ. The graphs of K 1 (Z 1 ) and K 2 (Z 2 ) as a function of transaction cost parameter λ are provided in Figure 2 and Figure 3, respectively K 1 (Z 1 ) λ Figure 2: K 1 (Z 1 ) when 0 < λ < 5% If λ increases, i.e., λ (recall Z 1 = (1 λ)(1 + R) (1 + r) and Z 2 = (1 λ)(r r)), both Z 1 and Z 2 will decrease (Z 1 and Z 2 ). Then immediately, we obtain F Z1, S Z1, F Z2, and S Z2, which, by definitions (9) and (11), imply that g 1 (Z 1 ), l 1 (Z 1 ), g 2 (Z 2 ), and l 2 (Z 2 ). All these results together suggest that K 1 (Z 1 ) and K 2 (Z 2 ), which are confirmed by Figure 2 and Figure 3. From Figures 2 and 3, we observe that 1 < K 1 (Z 1 ) < 2.25 and K 2 (Z 2 ) < 1 for all λ (0, 5%). In Tversky and Kahneman (1992), k is estimated to be 2.25, then Scenario 5(a) in Theorem 4.3 holds, and hence we obtain the optimal investment θ = 0. 29

30 K 2 (Z 2 ) λ Figure 3: K 2 (Z 2 ) when 0 < λ < 5% In this numerical example, the time window is chosen as one week and we have a bear market after the financial crisis of during the selected period; hence the difference between investment returns R(ω) r is small for most states ω Ω. With a longer time window and/or a better market performance, R r will increase, resulting in the increase of Z 1 and Z 2. Hence, we infer K 1 (Z 1 ) will be greater than 2.25 at certain model/market conditions when transaction costs are small. On the other hand, despite K 2 (Z 2 ) is an increasing function of λ (then a decreasing function of R r), K 2 (Z 2 ) is less sensitive to the change of λ or R r comparing to K 1 (Z 1 ). Therefore, in a bull market, we may have the case max{k 1 (Z 1 ), K 2 (Z 2 )} = K 1 (Z 1 ) > k for small λ, which corresponds to Case (5e) in Theorem 4.3, and then θ = θ M. The economic interpretation for this scenario is that CPT investors should buy the risky asset as much as they can in a very good economy. For example, if we assume the price process of the risky asset is given by a geometric Brownian Motion with drift 15% and volatility 20% and the risk-free interest rate is r = 5%. In addition, we select λ = 1% and T = 1 year. We find K 1 (Z 1 ) = > k = 2.25 and K 2 (Z 2 ) = , and thus θ = θ M. Clearly, if λ is large enough (e.g., λ 1), the optimal investment will be 0. In scenarios when K 1 (Z 1 ) > k for small λ, the impact of transaction costs on the optimal investment θ is dramatic, because θ = θ M if λ is less than a critical threshold, but θ = 0 if λ is greater than the threshold. 30

31 6.3 The Case of α < β We next study the case of α < β when ln(1 + R) is normally distributed. In this case, the optimal investment strategy is given by (5f) of Theorem 4.3. We investigate the impact of the utility parameters, α and β, on the optimal investment strategy. In this particular study, we assume the investment constraint is not binding, and hence, θ = arg max J(θ), {θ 1, θ 2 } where θ 1 and θ 2 are given by (10) and (12), respectively. The proof of Theorem 4.3 provides conditions when θ = θ 1 or θ 2, see Remark 4.2 for details θ 1 θ 2 θ * β Figure 4: α = 0.88, 0.88 < β < 1, and λ = 1% First, we fix α = 0.88, and calculate θ 1 and θ 2 as functions of β, where 0.88 < β < 1. The transaction cost parameter λ is chosen at 1%. In Figure 4, the line marked in circle coincides with the dashed line, i.e., θ = θ 2, implying that the optimal strategy is to short the risky asset. Furthermore, we observe that θ 1 is an increasing function of β, 7 but θ 2 is a decreasing function of β. Therefore, the optimal investment (in absolute amount) increases when β increases (i.e., CPT investors become less risk averse toward losses 8 ). 7 The increasing property of θ 1 with respect to β is not that noticeable in Figure 4, but is clearly supported by numerical values. 8 Recall the loss utility u ( x) = k( x) β, x < 0. 31

32 Next, we fix β = 0.88 and λ = 1%, and consider α (0.6, 0.88). By following similar numerical calculations as in the previous study, we draw the graphs in Figure 5. Comparing with the findings from Figure 4, we obtain exactly opposite results regarding monotonicity. Namely, θ 1 is a decreasing function of α and θ 2 is an increasing function of α. As before, we still have θ = θ 2. Therefore, this study shows that the optimal investment (in absolute amount) decreases as α increases (meaning CPT investors become less risk averse towards gains) θ 1-1 θ 2 θ * α Figure 5: 0.6 < α < 0.88, β = 0.88, and λ = 1% Lastly, we fix α = 0.8 and β = 0.88, and consider λ (0, 0.15%] (between 0 and 15 bps). The results in this case are drawn in Figure 6. Notice that we have θ = θ 1 only when transaction costs are small and θ = θ 2 otherwise. This result shows that transaction costs are crucial to the optimal investment strategy. Once the transaction cost parameter λ increases beyond a certain threshold (7 8 bps in the numerical example), the optimal investment strategy will shift from long position to short position in the risky asset. 32

33 θ 1 4 θ 2 θ * λ Conclusions Figure 6: α = 0.8, β = 0.88, and 0 < λ 0.15% Prospect theory was proposed in Kahneman and Tversky (1979), and further developed into cumulative prospect theory (CPT) in Tversky and Kahneman (1992). According to CPT, people evaluate uncertain outcomes by comparing them to a reference point, which separates all the outcomes into gains and losses based on the comparison. In addition, people s risk attitudes towards gains and losses are not universally risk averse. Instead, they exhibit fourfold patterns (see Tversky and Kahneman (1992)): risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability. The experimental studies challenge some fundamental axioms of expected utility theory (EUT), which, by far, is still the most popular criterion in economics and finance when it comes to decision making with uncertainty. In this paper, we consider a CPT investor in a single-period discretetime financial model with transaction costs. The investor seeks the optimal investment strategy that maximizes the prospect value of his/her final wealth. 33

34 The main objective of our work is to obtain explicit solutions to the optimal investment problem with transaction costs under CPT. We have successfully found the optimal investment in explicit form to this problem in two examples. We conduct an economic analysis to study the impact of transaction costs and risk aversion on the optimal investment. The results confirm that transaction costs play an important role in the optimal investment. There exist thresholds for the transaction cost parameter λ. In some cases, the optimal investment is 0 when λ is above a threshold. In other cases, there exists a threshold for λ which separates the optimal investment into buy strategies and sell strategies. We also observe that the optimal investment is affected by the investor s risk aversion parameters α and β. If investors become less risk averse towards gains (corresponding to the increase of α) or more risk averse towards losses (corresponding to the decrease of β), they will spend less in the risky asset. References Barberis, N., and Huang, M., Stocks as lotteries: the implications of probability weighting for security prices. American Economic Review 98(5): Bernard, C., and Ghossoub, M., Static portfolio choice under cumulative prospect theory. Mathematics and Financial Economics 2(4): Bernoulli, D., Exposition of a new theory on the measurement of risk (translated by Louise Summer). Econometrica 22: Carassus, L., and Rásonyi, M., On optimal investment for a behavioral investor in multiperiod incomplete market models. Mathematical Finance 25(1): Carlier, G., and Dana, R., Optimal demand for contingent claims when agents have law invariant utilities. Mathematical Finance 21(2): Davis, M., and Norman, A., Portfolio selection with transaction costs. Mathematics of Operations Research 15(4): He, X.D., and Zhou, X.Y., Portfolio choice under cumulative prospect theory: an analytical treatment. Management Science 57(2):